Constraints on Gauged B-3L_tau and Related Theories

We consider extensions of the Standard Model with an extra U(1) gauge boson which couples to B-(alpha L_e + beta L_mu + gamma L_tau) with alpha + beta + gamma = 3. We show that the extra gauge boson necessarily mixes with the Z leading to potentially significant corrections to the Z-f-fbar vertex. The constraints on the size of this correction imposed by the Z-pole data from LEP and SLD are derived.